Conifold Research Articles

Open Gromov-Witten invariants and SYZ under local conifold transitions

For a local Calabi-Yau manifold which is a smoothing of toric Gorenstein singularity, this paper computes the open Gromov-Witten invariants of a generic fiber of the special Lagrangian fibration constructed by Gross and thereby constructs its SYZ mirror. Moreover it derives a relation for the generating function of open Gromov-Witten invariants of the local Calabi-Yau manifold and that of its conifold transition.

Conifold transitions via affine geometry and mirror symmetry

Mirror symmetry of Calabi-Yau manifolds can be understood via a Legendre duality between a pair of certain affine manifolds with singularities called tropical manifolds. In this article, we study conifold transitions from the point of view of Gross and Siebert. We introduce the notions of tropical nodal singularity, tropical conifolds, tropical resolutions and smoothings. We interpret known global obstructions to the complex smoothing and symplectic small resolution of compact nodal Calabi-Yaus in terms of certain tropical $2$-cycles containing the nodes in their associated tropical conifolds. We prove that the existence of such cycles implies the simultaneous vanishing of the obstruction to smoothing the original Calabi-Yau \emph{and} to resolving its mirror. We formulate a conjecture suggesting that the existence of these cycles should imply that the tropical conifold can be resolved and its mirror can be smoothed, thus showing that the mirror of the resolution is a smoothing. We partially prove the conjecture for certain configurations of nodes and for some interesting examples.

Intersection spaces, perverse sheaves and type IIB string theory

The method of intersection spaces associates rational Poincar\'e complexes to singular stratified spaces. For a conifold transition, the resulting cohomology theory yields the correct count of all present massless 3-branes in type IIB string theory, while intersection cohomology yields the correct count of massless 2-branes in type IIA theory. For complex projective hypersurfaces with an isolated singularity, we show that the cohomology of intersection spaces is the hypercohomology of a perverse sheaf, the intersection space complex, on the hypersurface. Moreover, the intersection space complex underlies a mixed Hodge module, so its hypercohomology groups carry canonical mixed Hodge structures. For a large class of singularities, e.g., weighted homogeneous ones, global Poincar\'e duality is induced by a more refined Verdier self-duality isomorphism for this perverse sheaf. For such singularities, we prove furthermore that the pushforward of the constant sheaf of a nearby smooth deformation under the specialization map to the singular space splits off the intersection space complex as a direct summand. The complementary summand is the contribution of the singularity. Thus, we obtain for such hypersurfaces a mirror statement of the Beilinson-Bernstein-Deligne decomposition of the pushforward of the constant sheaf under an algebraic resolution map into the intersection sheaf plus contributions from the singularities.

CONIFOLD TRANSITIONS FOR COMPLETE INTERSECTION CALABI–YAU THREEFOLDS IN PRODUCTS OF PROJECTIVE SPACES

We prove that a generic complete intersection Calabi–Yau threefold defined by sections of ample line bundles on a product of projective spaces admits a conifold transition to a connected sum of S3 × S3. In this manner, we obtain complex structures with trivial canonical bundles on some connected sums of S3 × S3. This construction is an analogue of that made by Friedman, Lu and Tian who used quintics in ℙ4.

Completing the web of ℤ3‐quotients of complete intersection Calabi‐Yau manifolds

AbstractWe complete the study [1] of smooth ℤ3‐quotients of complete intersection Calabi‐Yau threefolds by discussing the six new manifolds that admit free ℤ3 actions that were discovered in [2]. These manifolds were missed in [1] and complete the web of smooth ℤ3‐quotients in a nice way. We discuss the transitions between these manifolds and include also the other manifolds of the web. This leads to the conclusion that the web of ℤ3‐free quotients of complete intersection Calabi‐Yau threefolds is connected by conifold transitions.

New construction of complex manifold via conifold transition

For a generic anti-canonical hypersurface in each smooth toric Fano 4-fold with rank 2 Picard group, we prove there exist three isolated rational curves in it. Moreover, for all these 4-folds except one, the contractions of generic anti-canonical hypersurfaces along the three rational curves can be deformed to smooth threefolds which is diffeomorphic to connected sums of S3 × S3. In this manner, we obtain complex structures with trivial canonical bundles on some connected sums of S3 × S3. This construction is an analogue of that made by Friedman [On threefolds with trivial canonical bundle. In: Complex Geometry and Lie Theory, volume 53 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1991, 103–134], Lu and Tian [Complex structures on connected sums of S3 × S3. In: Manifolds and Geometry, Pisa, 1993, 284–293] who used only quintics in ℙ4.

Moduli stabilisation in early superstring cosmology

We study the cosmological evolution induced by the free energy of a string gas at finite temperature. We first consider the maximally supersymmetric heterotic string in the weak coupling regime in dimension D ≥ 4. The free energy develops local minima associated to some gauge symmetry enhancement induced by perturbative states. This can stabilize all heterotic moduli except the dilaton. By string-string S-dualities, it follows that the dual type I moduli can be stabilized by either non perturbative D-string states or perturbative open string states. We next consider the type II string compactified on Calabi-Yau three-folds, which has \U0001d4a9 = 2 supersymmetry in D = 4. The moduli space has singular loci where the Calabi-Yau space undergoes extremal transitions, accompanied with either conifold transitions or certain gauge symmetry enhancement. Also the free energy reaches local minima due to the emergence of extra non perturbative massless modes. Consequently, the type II moduli are attracted to these singular loci. Using the type II/heterotic string duality, we can stabilize the dual heterotic moduli, and in particular it is possible to stabilize the heterotic dilaton.

Existence of Hermitian–Yang–Mills metrics under conifold transitions

We first study the degeneration of a sequence of Hermitian-Yang-Mills metrics with respect to a sequence of balanced metrics on a Calabi-Yau threefold $\hat{X}$ that degenerates to the balanced metric constructed by Fu, Li, and Yau on the complement of finitely many (-1,-1)-curves in $\hat{X}$. Then under some assumptions we show the existence of Hermitian-Yang-Mills metrics on bundles over a family of threefolds $X_t$ with trivial canonical bundles obtained by performing conifold transitions on $\hat{X}$.

Hyperconifold transitions, mirror symmetry, and string theory

Multiply-connected Calabi–Yau threefolds are of particular interest for both string theorists and mathematicians. Recently it was pointed out that one of the generic degenerations of these spaces (occurring at codimension one in moduli space) is an isolated singularity which is a finite cyclic quotient of the conifold; these were called hyperconifolds. It was also shown that if the order of the quotient group is even, such singular varieties have projective crepant resolutions, which are therefore smooth Calabi–Yau manifolds. The resulting topological transitions were called hyperconifold transitions, and change the fundamental group as well as the Hodge numbers. Here Batyrevʼs construction of Calabi–Yau hypersurfaces in toric fourfolds is used to demonstrate that certain compact examples containing the remaining hyperconifolds — the Z 3 and Z 5 cases — also have Calabi–Yau resolutions. The mirrors of the resulting transitions are studied and it is found, surprisingly, that they are ordinary conifold transitions. These are the first examples of conifold transitions with mirrors which are more exotic extremal transitions. The new hyperconifold transitions are also used to construct a small number of new Calabi–Yau manifolds, with small Hodge numbers and fundamental group Z 3 or Z 5 . Finally, it is demonstrated that a hyperconifold is a physically sensible background in Type IIB string theory. In analogy to the conifold case, non-perturbative dynamics smooth the physical moduli space, such that hyperconifold transitions correspond to non-singular processes in the full theory.

Wall Crossing of D4/D2/D0 on the Conifold

Wall Crossing of D4/D2/D0 on the Conifold

New Calabi‐Yau manifolds with small Hodge numbers

AbstractIt is known that many Calabi‐Yau manifolds form a connected web. The question of whether all Calabi‐Yau manifolds form a single web depends on the degree of singularity that is permitted for the varieties that connect the distinct families of smooth manifolds. If only conifolds are allowed then, since shrinking two‐spheres and three‐spheres to points cannot affect the fundamental group, manifolds with different fundamental groups will form disconnected webs. We examine these webs for the tip of the distribution of Calabi‐Yau manifolds where the Hodge numbers (h11, h21) are both small. In the tip of the distribution the quotient manifolds play an important role. We generate via conifold transitions from these quotients a number of new manifolds. These include a manifold with χ = ‐6, that is an analogue of the χ = ‐6 manifold found by Yau, and manifolds with an attractive structure that may prove of interest for string phenomenology. We also examine the relation of some of these manifolds to the remarkable Gross‐Popescu manifolds that have Euler number zero.

Constructing new Calabi-Yau 3-folds and their mirrors via conifold transitions

We construct a surprisingly large class of new Calabi–Yau 3-folds $X$ with small Picard numbers and propose a construction of their mirrors $X^∗$ using smoothings of toric hypersurfaces with conifold singularities. These new examples are related to the previously known ones via conifold transitions. Our results generalize the mirror construction for Calabi– Yau complete intersections in Grassmannians and flag manifolds via toric degenerations. There exist exactly 198849 reflexive four-polytopes whose two-faces are only triangles or parallelograms of minimal volume. Every such polytope gives rise to a family of Calabi–Yau hypersurfaces with at worst conifold singularities. Using a criterion of Namikawa we found 30241 reflexive four-polytopes such that the corresponding Calabi–Yau hypersurfaces are smoothable by a flat deformation. In particular, we found 210 reflexive four-polytopes defining 68 topologically different Calabi–Yau 3-folds with $h_11 = 1$. We explain the mirror construction and compute several new Picard–Fuchs operators for the respective oneparameter families of mirror Calabi–Yau 3-folds.

The making of Calabi‐Yau spaces: Beyond toric hypersurfaces

AbstractWhile Calabi‐Yau hypersurfaces in toric ambient spaces provide a huge number of examples, theoretical considerations as well as applications to string phenomenology often suggest a broader perspective. With even the question of finiteness of diffeomorphism types of CY 3‐folds unsettled, an important idea is Reid's conjecture that the moduli spaces are connected by certain singular transitions. We summarize the results of our recent construction of a large class of new CY spaces with small Picard numbers and of their mirrors via conifold transitions and discuss the benefits of other approaches to interesting locations in the web that have been or should be pursued.

Singular symplectic flops and Ruan cohomology

In this paper, we study the symplectic geometry of singular conifolds of the finite group quotient W r = { ( x , y , z , t ) ∣ x y − z 2 r + t 2 = 0 } / μ r ( a , − a , 1 , 0 ) , r ≥ 1 , which we call orbi-conifolds. The related orbifold symplectic conifold transition and orbifold symplectic flops are constructed. Let X and Y be two symplectic orbifolds connected by such a flop. We study orbifold Gromov–Witten invariants of exceptional classes on X and Y and show that they have isomorphic Ruan cohomologies. Hence, we verify a conjecture of Ruan.

Primitive Contractions of Calabi–Yau Threefolds I

We construct examples of primitive contractions of Calabi–Yau threefolds with exceptional locus being ℙ1 × ℙ1, ℙ2, and smooth del Pezzo surfaces of degrees ≤ 5. We describe the images of these primitive contractions and find their smoothing families. In particular, we give a method to compute the Hodge numbers of a generic fiber of the smoothing familly of each Calabi–Yau threefold with one isolated singularity obtained after a primitive contraction of type II. As an application, we get examples of natural conifold transitions between some families of Calabi–Yau threefolds.

Conormal bundles to knots and the Gopakumar-Vafa conjecture

We offer a new construction of Lagrangian submanifolds for the Gopakumar-Vafa conjecture relating the Chern-Simons theory on the 3-sphere and the Gromov-Witten theory on the resolved conifold. Given a knot in the 3-sphere, its conormal bundle is perturbed to disconnect it from the zero-section and then pulled through the conifold transition. The construction produces totally real submanifolds of the resolved conifold that are Lagrangian in a perturbed symplectic structure and correspond to knots in a natural and explicit way. We prove that both the resolved conifold and the knot Lagrangians in it have bounded geometry, and that the moduli spaces of holomorphic curves ending on the Lagrangians are compact in the Gromov topology.

Geometric transitions and mixed Hodge structures

We formulate and prove a B-model disc level large N duality result for general conifold transitions between compact Calabi-Yau spaces using degenerations of Hodge structures.

Universal character and large N factorization in topological gauge/string theory

We establish a formula of the large N factorization of the modular S-matrix for the coupled representations in U ( N ) Chern–Simons theory. The formula was proposed by Aganagic, Neitzke and Vafa, based on computations involving the conifold transition. We present a more rigorous proof that relies on the universal character for rational representations and an expression of the modular S-matrix in terms of the specialization of characters.

The effects of inhomogeneities on the cosmology of type IIB conifold transitions

In this paper we examine the evolution of the effective field theory describing a conifold transition in type IIB string theory. Previous studies have considered such dynamics starting from the cosmological approximation of homogeneous fields, here we include the effects of inhomogeneities by using a real-time lattice field theory simulation. By including spatial variations we are able to simulate the effect of currents and the gauge fields which they source. We identify two different regimes where the inhomogeneities have opposite effects, one where they aid the system to complete the conifold transition and another where they hinder it. The existence of quantized fluxes in related systems has lead to the speculation that (unstable) string solutions could exist, using our simulations we give strong evidence that these string-like defects do not form.

COMPUTATIONS ON B-MODEL GEOMETRIC TRANSITIONS

We study geometric transitions on Calabi–Yau manifolds from the perspective of the B model at genus g=0, with h=1 holes. We first perform a trivial check of the physically predicted duality across the conifold transition. Afterwards, a more complicated case is worked out directly at the level of period integrals, without resorting to periods of the imbedded Riemann surface.